Periodic input estimation for linear periodic systems: Automotive engine applications

In this paper, we consider periodic linear systems driven by T₀-periodic signals that we desire to reconstruct. The systems under consideration are of the form , y=C(t)x, x∈Rⁿ, w∈R^m, y∈R^p, (m?p?n) where A(t), A₀(t), and C(t) are T₀-periodic matrices. The period T₀ is known. The T₀-periodic input signal w(t) is unknown but is assumed to admit a finite dimensional Fourier decomposition. Our contribution is a technique to estimate w from the measurements y. In both full state measurement and partial state measurement cases, we propose an efficient observer for the coefficients of the Fourier decomposition of w(t). The proposed techniques are particularly attractive for automotive engine applications where sampling time is short. In this situation, standard estimation techniques based on Kalman filters are often discarded (because of their relative high computational burden). Relevance of our approach is supported by two practical cases of application. Detailed convergence analysis is also provided. Under standard observability conditions, we prove asymptotic convergence when the tuning parameters are chosen sufficiently small.     

Existence of solutions to singular integral equations

Nonnegative solutions are established for singular integral equations of the form y(t) = h(t) + ∫^T₀ k(t, s)f(s, y(s)) ds for t ∈ [0, T]. Here f may be singular at y = 0.     

Stochastic adaptive pole-zero assignment with convergence analysis

The adaptive control u_n is designed for the stochastic system A(z)y_n+1 = B(z)u_n+C(z)w_n+1 with unknown constant matrix coefficients in the polynomials A(z), B(z) and C(z) in the shift-back operator with the purposes that (1) the unknown matrices are strongly consistently estimated and (2) the poles and zeros are replaced in such a way that the system itself is transferred to A⁰(z)y_n+1 = B⁰(z)u_n⁰+_n+1 with given A⁰(z), B⁰(z) and u_n⁰ so that the pole-zero assignment error {_n+1} is minimized. The problem of adaptive pole-zero assignment combined with tracking is also considered in this paper. Conditions used are imposed only on A(z), B(z) and C(z).     

Computer-aided design via optimization: A review

Many design problems, including control design problems, involve infinite dimensional constraints of the form φ(z, α) ≤ 0 for all α ? $\text{A}$, where α denotes time or frequency or a parameter vector. In other design problems, tuning or trimming of certain parameters, after manufacture of the system, is permitted; the corresponding constraint is that for each α in $\text{A}$ there exists a value τ (of the tuning parameter) in a permissible set T such that φ(z, α, t) < 0. Recent algorithms for solving design problems having such constraints are summarized.     

Weighted aggregation operators based on minimization

Minimization based aggregation operators A_g,D are discussed. Special attention is paid to weighting function g based cases related to some fixed dissimilarity function D. When D₂(x,y)=(x-y)², we recognize mixture operators and we recall some sufficient conditions for g ensuring the monotonicity of A_g,D. For D₁(x,y)=|x-y| and non-decreasing (non-increasing) g, A_g,D is shown to be the upper (lower) median whenever A_g,D is an aggregation operator.     

The existence of positive solutions of singular fractional boundary value problems

We discuss the existence of positive solutions for the singular fractional boundary value problem D^αu+f(t,u,u^′,D^μu)=0, u(0)=0, u^′(0)=u^′(1)=0, where 2<α<3, 0<μ<1. Here D^α is the standard Riemann-Liouville fractional derivative of order α, f is a Carathéodory function and f(t,x,y,z) is singular at the value 0 of its arguments x,y,z.     

Consistent initialization and perturbation analysis for abstract differential-algebraic equations

In this paper, we consider linear and time-invariant differential-algebraic equations (DAEs) Eẋ(t) = Ax(t) + f(t), x(0) = x ₀, where x(·) and f(·) are functions with values in Hilbert spaces X and Z. is assumed to be a bounded operator, whereas A is closed and defined on some dense subspace D(A). A transformation to a decoupling form leads to a DAE including an abstract boundary control system. Methods of infinite-dimensional linear systems theory can then be used to formulate sufficient criteria for an initial value being consistent with the given inhomogeneity. We will further derive estimates for the trajectory x(·) in dependence of the initial state x ₀ and the inhomogeneity f(·). In the theory of differential-algebraic equations, this is commonly known as perturbation analysis.     

Some remarks on Bianchi type-II, VIII, and IX models

Within the scope of anisotropic non-diagonal Bianchi type-II, VIII, and IX spacetimes it is shown that the off-diagonal components of the Einstein equations impose severe restrictions on the components of the energy-momentum tensor (EMT) in general. We begin with a metric with three functions of time, a(t), b(t), and c(t), and two spatial ones, f(z) and h(z). It is shown that if the EMT is assumed to be diagonal, and f = f(z), in all cosmological models in question b ∝ c, and the matter distribution is isotropic, i.e., T ₁ ¹ = T ₂ ² = T ₃ ³ . If f = const, which is a special case of BII models, the matter distribution may be anisotropic, but only the z axis is distinguished, and in this case b(t) is not necessarily proportional to c(t).     

A general method to derive implicit equations of curves and surfaces using interlineation and interflation of functions

A general method is proposed to derive equations of irregular curves O _D (x, y) = 0 and of irregular surfaces O _G (x, y, z) =0 in implicit form, where the functions O _D (x, y) and O _G (x, y, z) belong to a prescribed differentiability class. The method essentially involves interlineation and interflation of functions. An example is considered.     

Asymptotic behavior of nonoscillatory solutions of neutral difference equations

The authors consider the m^th-order neutral difference equation D_m(y(n) + p(n)y(n − k) + q(n)f(y(σ(n))) = e(n), where m ≥ 1, {p(n)}, {q(n)}, {e(n)}, and {a₁(n)}, {a₂(n)}, …, {a_m−1(n)} are real sequences, a_i(n) > 0 for i = 1,2,…, m−1, a_m(n) ≡ 1, D₀z(n) = y(n)+p(n)y(n − k), D_iz(n) = a_i(n)ΔD_i−1z(n) for i = 1,2, …, m, k is a positive integer, {σ(n)} → ∞ is a sequence of positive integers, and R → R is continuous with u f(u) > 0 for u ≠ 0. In the case where {q(n)} is allowed to oscillate, they obtain sufficient conditions for all bounded nonoscillatory solutions to converge to zero, and if {q(n)} is a nonnegative sequence, they establish sufficient conditions for all nonoscillatory solutions to converge to zero. Examples illustrating the results are included throughout the paper.     

Diameter variability of cycles and tori

The diameter of a graph is an important factor for communication as it determines the maximum communication delay between any pair of processors in a network. Graham and Harary [N. Graham, F. Harary, Changing and unchanging the diameter of a hypercube, Discrete Applied Mathematics 37/38 (1992) 265-274] studied how the diameter of hypercubes can be affected by increasing and decreasing edges. They concerned whether the diameter is changed or remains unchanged when the edges are increased or decreased. In this paper, we modify three measures proposed in Graham and Harary (1992) to include the extent of the change of the diameter. Let D^-k(G) is the least number of edges whose addition to G decreases the diameter by (at least) k, D⁺⁰(G) is the maximum number of edges whose deletion from G does not change the diameter, and D^+k(G) is the least number of edges whose deletion from G increases the diameter by (at least) k. In this paper, we find the values of D^-k(C_m), D^-1(T_m,n), D^-2(T_m,n), D⁺¹(T_m,n), and a lower bound for D⁺⁰(T_m,n) where C_m be a cycle with m vertices, T_m,n be a torus of size m by n.     

Computing monodromy via continuation methods on random Riemann surfaces

We consider a Riemann surface X defined by a polynomial f(x,y) of degree d, whose coefficients are chosen randomly. Hence, we can suppose that X is smooth, that the discriminant δ(x) of f has d(d−1) simple roots, Δ, and that δ(0)≠0, i.e. the corresponding fiber has d distinct points {y₁,…,y_d}. When we lift a loop 0∈γ⊂C−Δ by a continuation method, we get d paths in X connecting {y₁,…,y_d}, hence defining a permutation of that set. This is called monodromy.Here we present experimentations in Maple to get statistics on the distribution of transpositions corresponding to loops around each point of Δ. Multiplying families of “neighbor” transpositions, we construct permutations and the subgroups of the symmetric group they generate. This allows us to establish and study experimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups.Assuming that these two conjectures are true, we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization, under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions.     

Substitutions into propositional tautologies

We prove that there is a polynomial time substitution (y₁,…,yn):=g(x₁,…,xk) with k?n such that whenever the substitution instance A(g(x₁,…,xk)) of a 3DNF formula A(y₁,…,yn) has a short resolution proof it follows that A(y₁,…,yn) is a tautology. The qualification “short” depends on the parameters k and n.     

The panpositionable panconnectedness of augmented cubes

A graph G is panconnected if, for any two distinct vertices x and y of G, it contains an [x, y]-path of length l for each integer l satisfying d_G(x, y) ? l ? ∣V(G)∣ − 1, where d_G(x, y) denotes the distance between vertices x and y in G, and V(G) denotes the vertex set of G. For insight into the concept of panconnectedness, we propose a more refined property, namely panpositionable panconnectedness. Let x, y, and z be any three distinct vertices in a graph G. Then G is said to be panpositionably panconnected if for any d_G(x, z) ? l₁ ? ∣V(G)∣ − d_G(y, z) − 1, it contains a path P such that x is the beginning vertex of P, z is the (l₁ + 1)th vertex of P, and y is the (l₁ + l₂ + 1)th vertex of P for any integer l₂ satisfying d_G(y, z) ? l₂ ? ∣V(G)∣ − l₁ − 1. The augmented cube, proposed by Choudum and Sunitha [6] to be an enhancement of the n-cube Q_n, not only retains some attractive characteristics of Q_n but also possesses many distinguishing properties of which Q_n lacks. In this paper, we investigate the panpositionable panconnectedness with respect to the class of augmented cubes. As a consequence, many topological properties related to cycle and path embedding in augmented cubes, such as pancyclicity, panconnectedness, and panpositionable Hamiltonicity, can be drawn from our results.     

Strong stability of the mono-tubular heat exchanger equation with output feedback

We investigate asymptotic behavior of the C₀-semigroup T(t) associated with the mono-tubular heat exchanger equation with output feedback by a perturbation method. It is shown that T(t) is bounded if a constraint is satisfied by the parameters and the spatial distribution function. Further, applying the Arendt-Batty-Lyubich-Vu theorem, a criterion is established to judge strong stability of T(t).     

On the structure of generalized rough sets

In this paper we consider some fundamental properties of generalized rough sets induced by binary relations on algebras and show that

1.

Any reflexive binary relation determines a topology.

2.

If θ is a reflexive and symmetric relation on a set X, then O={A⊆X|θ_-(A)=A} is a topology such that A is open if and only if it is closed.

3.

Conversely, for every topological space (X,O) satisfying the condition that A is open if and only if it is closed, there exists a reflexive and symmetric relation R such that O={A⊆X|R_-(A)=A}.

4.

Let θ be an equivalence relation on X. For any pseudo ω-closed subset A of X, θ₋(A) is an ω-closed set if and only if ω(x, x, … , x) ∈ θ₋(A) for any x ∈ X.

Moreover we consider properties of generalized rough sets.     

Identification via Non-linear Filtering

Non-linear filtering problems are being recognized more arid more as significant. Generally, an exact solution is not available but must be approximated. However, this paper gives the exact non-linear filter associated with identifying a scalar stochastic dynamical system, dx(t)/dt=ax(t)+ξ(t), disturbed by gaussian white noise ξ(t) when the system state x(t) is observed in an additive gaussian white noise environment, i.e. z(t)=x(t)+η(t) is observed over the interval of time 0≤t≤T<∞. The plant parameter, a, is assumed to have the prior probability density P_A .(a). The solution, is obtained by giving the conditional probability density functional p(a\z(t), 0≤t≤T<∞). The minimum mean-square-error estimate, that is, the Bayes estimate or conditional expectation, is given along with the minimum-mean-square-error. Limiting cases are described.

This approach is a variation on the eigenfunction expansion schemes used in the stochastic signals in noise-detection problems by Helstrom and others. This approach circumvents solving various forms of the Fokker-Planck type of partial differential equations, for the conditional probability density. In general, the partial differential equations are highly coupled and non-linear.     

Positive solutions of singular Dirichlet and periodic boundary value problems

Let A and T be positive numbers. The singular differential equation (r(x)x′)′ = μq(t)f(t, x) is considered. Here r > 0 on (0, A] may be singular at x = 0, and f(t, x) ≤ 0 may be singular at x = 0 and x = A. Effective sufficient conditions imposed on r, μ, q, and f are given for the existence of a solution x to the above equation satisfying either the Dirichlet conditions x(0) = x(T) = 0 or the periodic conditions x(0) = x(T), x′(0) = x′(T), and, in addition, 0 < x < A on (0, T).     

On the centers of a language

The center of a language has been defined in [7, 8, 9] as the set of all words which have infinite right completions in the language. In this paper we extend this notion by taking into account also left and two-sided completions. Thus, for any language X, we consider the left center C_l(X), the right center C_r(X) and two different bilateral centers C₁(X) and C₂(X). Some properties of these centers are derived. In particular the main results of the paper give some general conditions under which C₁(X)=C₂(X)and C_r(C_l(X))=C_l(C_r(X)). These conditions deal with ‘strong’ and ‘weak’ iteration properties and ‘periodicity’ of a language.